9.1 Derivatives of Vector Functions; the Divergence

The gradient is a vector function of several variables. Such an entity is called
a vector field, and we can ask, how do we compute derivatives of such
things?

We will consider this question in three dimensions, where we can answer it
as follows.

Since a vector in three dimensions has three components, and each of these
will have partial derivatives in each of three directions, there are actually
nine partial derivatives of a vector field in any coordinate system.

All of these can be computed by the same rules used for computing partial derivatives
of scalar functions (often called scalar fields). Fortunately for us, there
are only two combinations of these that we usually encounter and that are worth
knowing about.

The first of these is the divergence, written as div v, or in
terms of the differential operator del,
which is the vector operator with components

Explicitly, it is the dot product of this differential operator with the vector
v

Being a dot product, it is a number and not a vector.

This is how the divergence is defined, and again it can be calculated
by straightforward differentiation, but we must also address the question: what
does it mean? Why is it of interest to us? How can we use it? How can we compute
it in other coordinate systems?
We will defer the answer to these questions until we have discussed integration,
since the answers are intimately related to that subject.

However we can use the rules of differentiation to deduce the
following useful statements:

The divergence of the sum of two vectors is the sum of their
individual divergences.

And the divergence of a function f multiplied by a vector v
is given as follows

Exercises:

9.1 Derive this equation.

9.2 Apply it to find the divergence of in
spherical coordinates. Recall that the vector
has components (x, y, z) in spherical coordinates.